Skip to content. | Skip to navigation

Personal tools

Log in

Sections

You are here: Home › Teaching › Grad. Math Econ › Problem Sets › PS5

Problem Set 5

Problem Set 5

Problem Set 5

Compute the rank of the following matrix: $\begin{displaymath}\left[ \begin{array}{cccc} 1 & 6 & -7 & 3\\ 1 & 9 & -6 & 4 \\ 1 & 3 & -8 & 4 \end{array}\right] \end{displaymath}$
Evaluate the determinants.
1. $\left[ \begin{array}{ccc} 1 & 3 & 2 \\ 8 & 4 & 0 \\ 2 & 1 & 2 \end{array}\right]$
2. $\left[ \begin{array}{ccc} 0 & a & 1 \\ 1 & 0 & b \\ c & 1 & 0 \end{array}\right]$
3. $\left[ \begin{array}{cccc} 1 & 1 & 1 & -1 \\ 1 & 1 & -1 & 1 \\ 1 & -1 & 1 & 1 \\ -1 & 1 & 1 & 1 \end{array}\right]$
Determine the values of for which the matrix

$\displaystyle \left[ \begin{array}{ccc} t & 1 & 0 \ 0 & t & 0 \ 0 & 0 & t+3 \end{array} \right]$

is invertible.
Let $\mathbf{A,B,C}$ be invertible matrices of the same order. Simplify the expressions
1. $\left( I+A\right) A^{-1}\left( I-A\right)$
2. $A\left( 3A^{-1}+4B^{-1}\right) B$
3. $\left( AB^{-1}C\right) ^{-1}$
Use Cramer's Rule to solve the following equation systems:
1. $\displaystyle 3x_{1}-2x_{2}$ $\displaystyle =$ $\displaystyle 11$
  
  $\displaystyle 2x_{1}+x_{2}$ $\displaystyle =$ $\displaystyle 12$
2. $\displaystyle 10x+y-2z$ $\displaystyle =$ $\displaystyle 30$
  
  $\displaystyle x-y$ $\displaystyle =$ $\displaystyle 1$
  
  $\displaystyle 3x-4y+z$ $\displaystyle =$ $\displaystyle 15$
Find the ranks of the following matrices .
1. $\left[ \begin{array}{cc} 3 & 0 \\ 5 & 1 \\ 9 & 1 \end{array}\right]$
2. $\left[ \begin{array}{cccc} 1 & 2 & 3 & 4 \\ 3 & 6 & 9 & 11 \end{array}\right]$
Let $\mathbf{Ax=b}$ be a linear system of equation in matrix form. Prove that if $\mathbf{x}_{1}$ and $\mathbf{x}_{2}$ are both solutions of the system, then so is $\lambda \mathbf{x}_{1}+\left( 1-\lambda \right) \mathbf{x}_{2}$ for every real number $\lambda .$ Use this fact to prove that a linear system of equations that is consistent has either one solution or infinitely many solutions. (For instance, it cannot have exactly 3 solutions.)
An orthogonal matrix is an invertible square matrix whose inverse is its transpose.
Show that

$\displaystyle \left[ \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \right]$ and $\displaystyle \frac{1}{\sqrt{2}}\left[ \begin{array}{cc} 1 & 1 \ 1 & -1 \end{array} \right]$

are orthogonal matrices.
Prove that the determinant of an orthogonal matrix is either or . Show that both cases can occur.
Let be constants such that $abc\neq 1.$ Use Cramer's rule to solve the system of equations

$\displaystyle ay+z$ $\displaystyle =$ $\displaystyle 1$

$\displaystyle x+bz$ $\displaystyle =$ $\displaystyle 1$

$\displaystyle cx+y$ $\displaystyle =$ $\displaystyle 1.$
Write the following quadratic form in matrix form: $5x_{1}^{2} - 10x_{1}x_{2} - x_{2}^{2}$
Determine the values of for which the symmetric matrix $\begin{displaymath}\left[ \begin{array}{cc} 2t & 2 \\ 2 & t \end{array}\right] \end{displaymath}$ is
1. positive definite
2. positive semidefinite but not positive definite
3. negative definite
4. negative semidefinite but not negative definite
5. none of the above
Determine the definiteness of the symmetric matrices
1. $\left[ \begin{array}{ccc} 3 & -1 & 1 \\ -1 & 1 & 2 \\ 1 & 2 & 9 \end{array}\right]$
2. $\left[ \begin{array}{ccc} 3 & -1 & 1 \\ -1 & 1 & 2 \\ 1 & 2 & 6 \end{array}\right]$
3. $\left[ \begin{array}{ccc} 1 & -1 & 0 \\ 1 & -2 & 1 \\ 0 & 1 & -1 \end{array}\right]$
Let $\mathbf{A}=\left( a_{ij}\right) _{nxn}$ be symmetric and positve semidefinite. Prove that $\mathbf{A}$ is positive definite if and only if $\left\vert\mathbf{A}\right\vert \neq 0$ .

About this document ...

Jennifer Anne Thacher 2008-09-27

Document Actions

Send this